András Bárdossy

Zusammenfassung

Der Beitrag präsentiert Ergebnisse eines Forschungsprojektes, welches sich mit der Untersuchung von Niederschlagsmodellen zur Erzeugung von synthetischen Niederschlagsdaten für stadthydrologische Anwendungen beschäftigt hat. Es werden zwei ausgewählte Niederschlagsmodelle vorgestellt und verglichen. Die Modelle wurden unter Verwendung der Daten von ungefähr 800 Niederschlagsstationen für ganz Deutschland regionalisiert und für 45 ausgewählte repräsentative Standorte über Statistiken von Niederschlag und simuliertem Abfluss in Kanalnetzen validiert. Die Ergebnisse zeigen die Überlegenheit der synthetischen Niederschläge im Vergleich zu einem Praxisszenario, welches immer die nächstgelegene verfügbare Station zur Datenlieferung verwendet. Die besten Ergebnisse liefert eine Mischung der Daten aus beiden Modellen, die zu einer Fehlerkompensation führt. Es werden Empfehlungen zur notwendigen Zeitreihenlänge für die praktische Bemessung gegeben. Die synthetischen Niederschläge stehen jetzt für beliebige Punkte in Deutschland auf einer Webplattform für die wasserwirtschaftliche Praxis zur Verfügung.

Zusammenfassung

Study region: The study region is Germany and two sub-regions in Germany, i.e. the state of Rhineland-Palatinate and the city of Reutlingen. Study focus: Opportunistic rainfall sensors, namely personal weather stations and commercial microwave links, together with rain gauge data from the German Weather Service, were used in different combinations to derive rainfall maps with a geostatistical interpolation framework for Germany. This kriging type framework considered the uncertainty of opportunistic sensors and the line structure of commercial microwave links. The resulting rainfall maps were compared to two gauge-adjusted radar products and evaluated to three reference gauge datasets in the respective study regions on both daily and hourly basis. New Hydrological Insights for the Region: The interpolated rainfall products from opportunistic sensors provided good agreement to the reference rain gauges. The dataset combinations including information from the opportunistic sensors performed best. The addition of rain gauges from the German Weather Service did not consistently lead to an improvement of the interpolated rainfall maps. On the country-wide, daily scale the interpolated rainfall maps performed well, but the gauge-adjusted radar products were closer to the reference. For the regional and local scale in Rhineland-Palatinate and Reutlingen with an hourly resolution, the interpolated rainfall maps outperformed the interpolated product from DWD rain gauges and showed a similar agreement to the reference as the radar products.

Zusammenfassung

Abstract The number of active gauges with open-data policy for discharge monitoring along rivers has decreased over the last decades. Therefore, spaceborne measurements are investigated as alternatives. Among different techniques for estimating river discharge from space, developing a rating curve between the ground-based discharge and spaceborne river water level or width is the most straightforward one. However, this does not always lead to successful results, since the river section morphology often cannot simply be modeled by a limited number of parameters. Moreover, such methods do not deliver a proper estimation of the discharge’s uncertainty as a result of the mismodeling and also the coarse assumptions made for the uncertainty of inputs. Here we propose a nonparametric model for estimating river discharge and its uncertainty from spaceborne river width measurements. The model employs a stochastic quantile mapping scheme by, iteratively: 1) generating realizations of river discharge and width time series using Monte Carlo simulation, 2) obtaining a collection of quantile mapping functions by matching all possible permutations of simulated river discharge and width quantile functions, 3) adjusting the measurement uncertainties according to the point cloud scatter. We validate our method over 14 different river reaches along the Niger, Congo, Po rivers and several river reaches in the Mississippi river basin. Our results show that the proposed algorithm can mitigate the effect of measurement noise and also possible mismodelling. Moreover, the proposed algorithm delivers a meaningful uncertainty for the estimated discharge and allows us to calibrate the error bars of in situ discharge measurements.

Zusammenfassung

In this study, the applicability of data from private weather stations (PWS) for precipitation interpolation was investigated. Due to unknown errors and biases in these observations, a two-step filter was developed that uses indicator correlations and event-based spatial precipitation patterns. The procedure was tested and cross validated for the state of Baden-Württemberg (Germany). The biggest improvement is achieved for the shortest time aggregations.

Zusammenfassung

Freshwater is distributed in a very non-uniform manner on the surface of the earth. Both spatial and temporal variability have a strong influence on water management. In this chapter both mean values and spatial and temporal variability measures of the main hydrological variables, precipitation (rain and snow), discharge, groundwater availability are mapped in order to enable specific evaluation of the regional water management requirements. Besides the natural variables socio-economic factors are also investigated, water availability is presented for the different countries of the world.

Zusammenfassung

Precipitation is one of the most important variables in the global climate system, which is almost surely affected by climate change. Changes in precipitation series might not only affect means, variances and extremes in a single location, but also the spatial and temporal scales. Simultaneous occurrences of precipitation anomalies on the multivariate scale are called unusual precipitation events. In this article, a newly developed method is introduced to identify the occurrence of unusual precipitation events in multivariate datasets. This method is based on a statistical approach using the outlyingness function with the consideration of precipitation values of zero. Several selected precipitation observation stations in Europe were used in this investigation to illustrate the methodology. The investigation was conducted at the different multi-site and temporal scales, with consideration of short time intervals (2–5-day sequences) and the spatial of precipitation series. The results show that the occurrence of unusual precipitation events has a dynamic behavior in the spatial and temporal scales. A high uncertainty of unusual precipitation events was identified at different scales. The statistical significance of the results was tested using a bootstrap-based methodology.

Zusammenfassung

The accuracy of quantitative precipitation estimation (QPE) over a given region and period is of vital importance across multiple domains and disciplines. However, due to the intricate temporospatial variability and the intermittent nature of precipitation, it is challenging to obtain QPE with adequate accuracy. This paper aims to simulate rainfall fields while honoring both the local constraints imposed by the point-wise rain gauge observations and the global constraints imposed by the field measurements obtained from weather radar. The conditional simulation method employed in this study is modified phase annealing (PA), which is practically an evolution from the traditional simulated annealing (SA). Yet unlike SA, which implements perturbations in the spatial field, PA implements perturbations in Fourier space, making it superior to SA in many respects. PA is developed in two ways. First, taking advantage of the global characteristic of PA, the method is only used to deal with global constraints, and the local ones are handed over to residual kriging. Second, except for the system temperature, the number of perturbed phases is also annealed during the simulation process, making the influence of the perturbation more global at initial phases and decreasing the global impact of the perturbation gradually as the number of perturbed phases decreases. The proposed method is used to simulate the rainfall field for a 30 min event using different scenarios: with and without integrating the uncertainty of the radar-indicated rainfall pattern and with different objective functions.

Zusammenfassung

Abstract Observed trends, theory and modelling results all suggest increases in future extreme precipitation due to climate warming. The largest increases are expected in short-duration events with less than a day. Relatively few previous studies have focused specifically on the projection of sub-daily precipitation extremes. In this study, a statistical downscaling method based on circulation patterns (CPs) is developed to project site-specific extreme hourly precipitation over the United Kingdom. First, a CP-classification categorizes extreme hourly precipitation events based on the underlying atmospheric pressure conditions on each day. An analogue day method is then used to find for each future day the most similar day in the past by comparing the predictor values of daily precipitation and temperature simulated by regional climate models (RCMs) with observations conditioned on different CPs and seasons. Finally, the maximum hourly precipitation records on the most similar days are extracted and perturbed using precipitation duration-temperature relationships. The applied statistical downscaling method is a combination of the analogue and the regression-based method. It is found that the statistical downscaling method is able to reproduce observed extreme hourly precipitation. In terms of future changes under a warmer climate, it is shown that increases in extreme hourly precipitation can be as high as 112\% but are highly variable depending on the rainfall stations, the future time periods, the emission scenarios, and the different RCM runs.

Zusammenfassung

A lumped version of the conceptual HBV–IWS version (Hydrologiska Byrans Vattenbalansavdelning) hydrological model was applied on the Black Volta Catchment in the West African sub-region (a data-scarce catchment) to investigate the performance of the model in the study area. In assessing the robustness of the model and how well it performs in the study area, different optimization methods were used. The Robust Parameter Estimation (ROPE) algorithm was used to generate and calibrate 10,000 best parameter sets in a computer framework with initially set model parameter ranges. The model performed well with an average NS of 0.75 and 0.6 for calibration and validation respectively for subcatchments Lawra and Chache. The performance also improved when a higher weight was given to the low-flows during the optimization. Results also show that the model reacts well to precipitation signals. For a proper management of the water resources and hydropower in the West Africa sub-region and for good policy decisions and investments in water related projects, a robust hydrological model would be essential and crucial for the Black Volta Catchment. The results of this study will be an essential input to a further study in assessing the effects of climate change on the hydrology of the West African sub-region.

Zusammenfassung

With modern data acquisition technology, Electrical Resistivity Tomography (ERT) surveys can be deployed easily and run cost-efficiently over large areas and over long periods. Inversion of the electric potential data to image the subsurface resistivity distribution is a key step in the workflow. In contrast to deterministic inversion commonly used in ERT, geostatistical inversion methods construct a variety of possible realizations of resistivity distributions and allow for a better interpretation of results and a quantification of the uncertainty. In this paper we present a geostatistical approach for the inversion of ERT survey data based on Random Mixing. The method of Random Mixing had been developed and tested for inverse groundwater modelling and is applied here to ERT as an example of geophysical inversion for the first time. Random Mixing is based on the idea of constructing realizations of the unknown electric conductivity field by combining random fields representing the spatial correlation of conductivity such that direct measurements of conductivity if available and indirect dipole measurement of electric potentials at electrodes are matched. Implementation is based on the three-dimensional finite element method to solve the forward model where properties in a core region are updated through the Random Mixing iteration. To reduce computational costs for the evaluation of the misfit cost function one-dimensional interpolation is applied. The results of Random Mixing for a synthetic example and a field survey inversion are discussed and compared with deterministic inversion results.

Zusammenfassung

The application of hydrological models in data-scarce catchments is usually limited by the amount of available data. It is of great significance to investigate the impacts of data quantity and quality on model calibration—as well as to further improve the understanding of the effective estimation of robust model parameters. How to make adequate utilization of external information to identify model parameters of data-scarce catchments is also worthy of further exploration. HBV (Hydrologiska Byråns Vattenbalansavdelning) models was used to simulate streamflow at 15 catchments using input data of different lengths. The transferability of all calibrated model parameters was evaluated for two validation periods. A simultaneous calibration approach was proposed for data-scarce catchment by using data from the catchment with minimal spatial proximity. The results indicate that the transferability of model parameters increases with the increase of data used for calibration. The sensitivity of data length in calibration varies between the study catchments, while flood events show the key impacts on surface runoff parameters. In general, ten-year data are relatively sufficient to obtain robust parameters. For data-scarce catchments, simultaneous calibration with neighboring catchment may yield more reliable parameters than only using the limited data.

Zusammenfassung

Rain gauges are the foundation in hydrology to collect rainfall data, however gauge observations alone are limited at representing the complete rainfall distribution. On the other hand, weather radar can provide a nearly complete spatial distribution of rainfall qualitatively at high temporal and spatial resolution, yet concerns about the biases in the precipitation estimates hamper the direct use of radar data in hydrological applications. Thus the focus of this study is to derive a scheme to merge radar precipitation estimates and the contemporaneous point-wise rain gauge observations. It is common practice to construct radar-gauge pairs by assuming the vertical and instant falling of the hydrometeors from aloft onto the ground. On many occasions, the assumption is invalid, as it ignores a significant fact that the hydrometeors are very likely to be laterally advected in the air by the wind while descending, especially from a large height and/or with the existence of snow. The laterally displacement of rainfall will result in a great discrepancy of radar and gauge data. Concerning these issues, a method to quantify the displacement of rainfall is proposed and the results of the quantification are integrated into much improved surface precipitation estimates.

Zusammenfassung

Soil moisture is a key state variable in terrestrial water cycles, which links various land surface and hydrological processes. Owing to the significant spatiotemporal variability in soil moisture, collecting sufficient soil moisture data for relevant studies can be an astonishingly difficult task. Here, a statistical method based on the concept of depth functions was used to define Critical Events (CE) of soil moisture, which was hypothesized to contain disproportionally more system information on soil moisture dynamics. To test the feasibility of applying the CE concept for quantifying soil moisture dynamics, a long-term soil moisture dataset was retrieved from the Automated Weather Data Network (AWDN) located in the continental United States. The method of Temporal Stability Analysis (TSA) was adopted to examine the information embedded in soil moisture data that were collected under different conditions. The results showed that similar information on the relative as well as absolute wetness conditions at the AWDN sites was extracted from the entire time series and CEs of soil moisture, the latter of which contained much less soil moisture observations. Finally, the field data revealed that the occurrence of CEs of soil moisture in the study area was mainly affected by soil depth and interannual variability in precipitation. The number of CEs of soil moisture tended to be larger at deeper soil depths as well as in either dry or wet years than in normal years, suggesting that more soil moisture measurements under those conditions were needed to provide adequate information on soil moisture systems.

Zusammenfassung

Abstract A wide variety of processes controls the time of occurrence, duration, extent, and severity of river floods. Classifying flood events by their causative processes may assist in enhancing the accuracy of local and regional flood frequency estimates and support the detection and interpretation of any changes in flood occurrence and magnitudes. This paper provides a critical review of existing causative classifications of instrumental and preinstrumental series of flood events, discusses their validity and applications, and identifies opportunities for moving toward more comprehensive approaches. So far no unified definition of causative mechanisms of flood events exists. Existing frameworks for classification of instrumental and preinstrumental series of flood events adopt different perspectives: hydroclimatic (large-scale circulation patterns and atmospheric state at the time of the event), hydrological (catchment scale precipitation patterns and antecedent catchment state), and hydrograph-based (indirectly considering generating mechanisms through their effects on hydrograph characteristics). All of these approaches intend to capture the flood generating mechanisms and are useful for characterizing the flood processes at various spatial and temporal scales. However, uncertainty analyses with respect to indicators, classification methods, and data to assess the robustness of the classification are rarely performed which limits the transferability across different geographic regions. It is argued that more rigorous testing is needed. There are opportunities for extending classification methods to include indicators of space–time dynamics of rainfall, antecedent wetness, and routing effects, which will make the classification schemes even more useful for understanding and estimating floods. This article is categorized under: Science of Water > Water Extremes Science of Water > Hydrological Processes Science of Water > Methods

Zusammenfassung

Persistence in time series of daily weather pattern (WP) classifications can provide useful information such as on the memory of broad-scale atmospheric circulation. Despite this, research of WP persistence has lagged behind that exploring their frequencies of occurrence. We develop two methods for identifying persistence in a 167-year time series of WPs defined over the North Atlantic–European domain. The first is an empirical counting technique used to find periods of persistence among sets of WPs, with the definition of persistence more relaxed than just consecutive occurrences. We then condition this method on the driest WPs to see if persistence can be used to identify historical drought. The second method uses a Markov model to assess if WP transition probabilities change when conditioned on information up to 20 days prior, without the need for estimating the large number of parameters usually required for high-order Markov chains. Results are compared with a benchmark ensemble of synthetic time series generated using first-order transition probabilities. We show that there were multi-month periods when small sets of WPs dominated, and some of these periods coincided with notable meteorological events, including droughts and storms, such as the mid-1990s drought in northern England and the Burn's Day Storm over southern Scotland in 1990. Some WPs also behave as “attractors,” showing increased probability of reoccurrence despite other WPs occurring in-between. However, we find no link between the persistence statistics of each WP and their flow characteristics, except for those featuring an easterly flow over the United Kingdom, which are among the most persistent. The benchmark simulation ensemble is unable to reproduce many of the key persistence statistics of the observations, confirming that the persistence is a physical phenomenon. Finally, we discuss the potential processes underpinning WP persistence, such as the effects of large-scale circulation patterns and land-surface feedbacks.

Zusammenfassung

Information on extreme precipitation is essential to managing weather-related risks and designing hydraulic structures. Research attention to frequency analyses based on remotely sensed precipitation datasets, such as those obtained from weather radars and satellites, has been rapidly increasing owing to their potential to provide information for ungauged regions worldwide. Together with the ability to measure the areal scale directly, these analyses promise to overcome the sampling limitations of traditional methods based on rain gauges. This focused review of the literature depicts the state of the art after a decade of efforts, and identifies the crucial gaps in knowledge and methodology that currently hinder the quantitative use of remotely sensed datasets in water resources system design and operation. It concludes by highlighting a set of research directions promising immediate impact with regard to the separation of the sources of uncertainty currently affecting applications based on remotely sensed datasets, the development of statistical methods that can cope with the peculiar characteristics of these datasets, and the improvement of validation methods. Important gains in knowledge are expected from the explicit inclusion of the areal dimension in the analyses and from the fine-scale investigation of extreme precipitation climatology.

Zusammenfassung

The widely applied geostatistical interpolation methods of ordinary kriging (OK) or external drift kriging (EDK) interpolate the variable of interest to the unknown location, providing a linear estimator and an estimation variance as measure of uncertainty. The methods implicitly pose the assumption of Gaussianity on the observations, which is not given for many variables. The resulting “best linear and unbiased estimator” from the subsequent interpolation optimizes the mean error over many realizations for the entire spatial domain and, therefore, allows a systematic under-(over-)estimation of the variable in regions of relatively high (low) observations. In case of a variable with observed time series, the spatial marginal distributions are estimated separately for one time step after the other, and the errors from the interpolations might accumulate over time in regions of relatively extreme observations. Therefore, we propose the interpolation method of quantile kriging (QK) with a two-step procedure prior to interpolation: we firstly estimate distributions of the variable over time at the observation locations and then estimate the marginal distributions over space for every given time step. For this purpose, a distribution function is selected and fitted to the observed time series at every observation location, thus converting the variable into quantiles and defining parameters. At a given time step, the quantiles from all observation locations are then transformed into a Gaussian-distributed variable by a 2-fold quantile–quantile transformation with the beta- and normal-distribution function. The spatio-temporal description of the proposed method accommodates skewed marginal distributions and resolves the spatial non-stationarity of the original variable. The Gaussian-distributed variable and the distribution parameters are now interpolated by OK and EDK. At the unknown location, the resulting outcomes are reconverted back into the estimator and the estimation variance of the original variable. As a summary, QK newly incorporates information from the temporal axis for its spatial marginal distribution and subsequent interpolation and, therefore, could be interpreted as a space–time version of probability kriging. In this study, QK is applied for the variable of observed monthly precipitation from raingauges in South Africa. The estimators and estimation variances from the interpolation are compared to the respective outcomes from OK and EDK. The cross-validations show that QK improves the estimator and the estimation variance for most of the selected objective functions. QK further enables the reduction of the temporal bias at locations of extreme observations. The performance of QK, however, declines when many zero-value observations are present in the input data. It is further revealed that QK relates the magnitude of its estimator with the magnitude of the respective estimation variance as opposed to the traditional methods of OK and EDK, whose estimation variances do only depend on the spatial configuration of the observation locations and the model settings.

Zusammenfassung

Geostatistical inverse modeling problems can potentially be very high-dimensional and computationally expensive. Using the Random Mixing approach the dimensionality can be reduced, and any optimization algorithm can be applied to solve the inverse problem in this lower dimensional space. In order to reduce computational costs the standard optimization is replaced by a sequence of one-dimensional optimizations. In this one-dimensional space a simplified calculation of the objective function using Whittaker–Shannon interpolation is carried out. This procedure requires a significantly reduced number of forward model runs and it also guarantees monotonic convergence of the objective function. A synthetic and a real world example will be used to demonstrate the procedure and to evaluate the quality of the Whittaker–Shannon interpolation in comparison to the actual objective functions.

Zusammenfassung

Rainfall is the most important input for rainfall–runoff models. It is usually measured at specific sites on a daily or sub-daily timescale and requires interpolation for further application. This study aims to evaluate whether a higher temporal and spatial resolution of rainfall can lead to improved model performance. Four different gridded hourly and daily rainfall datasets with a spatial resolution of 1 km × 1 km for the state of Baden-Württemberg in Germany were constructed using a combination of data from a dense network of daily rainfall stations and a less dense network of sub-daily stations. Lumped and spatially distributed HBV models were used to investigate the sensitivity of model performance to the spatial resolution of rainfall. The four different rainfall datasets were used to drive both lumped and distributed HBV models to simulate daily discharges in four catchments. The main findings include that (1) a higher temporal resolution of rainfall improves the model performance if the station density is high; (2) a combination of observed high temporal resolution observations with disaggregated daily rainfall leads to further improvement in the tested models; and (3) for the present research, the increase in spatial resolution improves the performance of the model insubstantially or only marginally in most of the study catchments.

Zusammenfassung

Precipitation plays an important role in any hydrological analysis. The aim of this study is to empirically investigate the behaviour of spatial dependence of precipitation fields. This would determine whether the Gaussian assumption is fulfilled in regard to symmetric spatial dependence structure between low and high precipitation values. An asymmetry function that can incorporate zero precipitation amounts is introduced on the basis of empirical bivariate copulas. The asymmetry function is calculated by integrating the empirical bivariate copula density in the upper right and the lower left parts for any given quantile thresholds. Zero precipitation amounts are handled as latent variables and the thresholds, therefore, are set to be bigger than probability of zero. The Gaussian simulations based testing is applied for determining the degree of uncertainty. The empirical bivariate copulas are constructed using the concept of regionalized variables in spatial random process with a given separating distance. For any selected time interval, the precipitation over the region of interest is assumed to be a single realization of a spatially stationary random process. The investigations are conducted in Singapore and Bavaria. To take temporal characteristics and seasons into account, precipitation occurrences with different time scales for different seasons are analysed. The empirical evidence proves that the precipitation events tend to follow the positive asymmetric spatial dependence structure, particularly at a short separating distance. This implies that precipitation amounts with higher intensities tends to be more spatially correlated than that with lower intensities because precipitation occurrences tend to occur in a clustered manner. Consequently, spatial precipitation models that are based on the symmetric Gaussian dependence could result in an underestimation of the spatial extent of actual precipitation extremes. Moreover, this pattern is higher for smaller time scales (hourly to daily).

Zusammenfassung

Simulated annealing (SA) is a popular geostatistical simulation method as it provides great flexibility. In this paper possible problems of conditioning its realizations are discussed. A statistical test to recognize whether the observations are well embedded in their simulated neighborhood or not is developed. A new simulated annealing method, phase annealing (PA), is presented which makes it possible to avoid poor embedding of observations. PA is based on the Fourier representation of the spatial field. Instead of the individual pixel values, phases corresponding to different Fourier components are modified (i.e. shifted) in order to match prescribed statistics. The method treats neighborhoods together and thus avoids singularities at observation locations. It is faster than SA and can be used for the simulation of high resolution fields. Examples demonstrate the applicability of the method.

Zusammenfassung

Often, the primary variable of interest can be measured only rarely and/or indirectly. Hence, it is desired to use secondary correlated data to improve the estimation of the primary variable. This paper extends the Gaussian copula-based geostatistical approach for interpolation to include both real-valued primary and secondary data. Even more, multiple types of real-valued secondary data and categorical exhaustive secondary information were incorporated and the associated benefits analysed. The performance of the proposed inclusion of secondary data and information was compared to benchmark geostatistical methods (Ordinary Kriging, co-Kriging, external drift Kriging, and copula-based interpolation of the primary variable alone) by cross-validation and split sampling, with varying compositions of primary and secondary variables. The proposed method was tested for several groundwater quality parameters and the results show that a) the joint copula-based method outperforms co-Kriging in all quality measures; (b) physically impossible estimates (i.e., negative concentrations) never occur; (c) copula-based methods outperform Kriging methods in the quantification of uncertainty and result in more realistic spatial patterns of uncertainty. Joint methods lead to better results in cases when the primary variable is available at fewer locations than the secondary variable (i.e., undersampled case) and when the variables are meaningfully related. In copula-based spatial dependence models that are constructed in probability space, detrimental effects of non-normal (here: bi-modal) marginal distributions on the joint distribution and dissimilar measurement scales of the primary and secondary variables are prevented. The proposed method offers an efficient and non-linear alternative to co-Kriging.

Zusammenfassung

Records of precipitation extremes are essential for hydrological design. In urban hydrology, intensity–duration–frequency curves are typically estimated from observation records. However, conventional approaches seldom consider the areal extent of events. If they do, duration-dependent area reduction factors are used, but precipitation is measured at only a few locations. Due to the high spatial variability of precipitation, it is relatively unlikely that a gauged observation network will capture the extremes that occur during a precipitation event. Therefore, the area reduction approach cannot be regarded as the reduction of an observed maximum. To investigate precipitation extremes, spatial aspects need to be considered using different approaches. Here, we both address the conventional practice of area reduction and consider a within-area chance of increased precipitation, defined as the maximum precipitation intensity observed in a cluster within a selected domain. The results show that (1) the risk of urban flooding is routinely underestimated in current design practice, and (2) traditional calculations underestimate extremes by as much as 30–50\%. We show how they can be revised sensibly.

Zusammenfassung

The Multiplicative Discrete Random Cascade (MDRC) class of model is used to temporally disaggregate rainfall volumes through multiplying the volumes by random weights, which is repeated through multiple disaggregation levels. The model development involves the identification of probability density functions from which to sample the weights. The parameters of the probability density functions are known to be dependent on the rainfall volume. This paper characterises the volume dependency over the scarcely observed extreme ranges of rainfall, introducing the concept of volume-bounded MDRC models. Probable maximum precipitation (PMP) estimates are used to define theoretically-based points and asymptotes to which the observation-based estimates of the MDRC model parameters are extrapolated. Alternative models are tested using a case study of rainfall data from Brisbane, Australia covering the period 1908 to 2015. The results show that moving from a baseline model with constant parameters to incorporating the volume dependency of the parameters is essential for acceptable performance in terms of the frequency and magnitude of modelled extremes. As well as providing better estimates of parameters at each disaggregation level, the volume dependency provides an in-built bias correction when moving from one level to the next. A further, relatively small performance gain is obtained by extrapolating the observed dependency to the theoretically-based bounds. The volume dependency of the parameters is found to be reasonably time-scaleable, providing opportunity for advances in the generalisation of MDRC models. Sensitivity analysis shows that the subjectivities and uncertainties in the modelling procedure have mixed effects on the performance. A principal uncertainty, to which the results are sensitive, is the PMP estimate. Therefore, in applications of the bounded approach, the PMP should ideally be described by a probability distribution function.

Zusammenfassung

The estimation of an appropriate variogram is a crucial step toward the description of spatial dependence, the geostatistical interpolation of environmental variables, and the subsequent hydrological engineering. The classical variogram in the literature ideally necessitates a normal distribution of the variable and is not robust against outliers within the data. These presumptions are hardly given under empirical conditions and, therefore, a new estimation method is proposed for the variogram. The new method is based on the description of spatial dependence by the robust rank coefficient τ and generalizes the method from the Gaussian to the general case of empirical distributions. The conversion of the robust estimate using a Monte-Carlo simulation and subsequent quantile-quantile transformation with the empirical marginal distribution performs the generalization. Monthly precipitation data from South Africa serve as the variable and were artificially contaminated with outliers. The effects on the variogram and subsequent geostatistical interpolation were investigated for the proposed, classical, and four existing robust variogram models in this comparative study. The investigation revealed that the proposed variogram describes a distinct spatial dependence structure under empirical conditions, which is robust against outliers. The cross validation of the linear estimator demonstrates that the proposed variogram tends to improve the bias and spread of the resulting error distribution, and hence the quality of the geostatistical interpolation.

Zusammenfassung

This paper demonstrates a maximum likelihood (ML)-based approach to derive representative (“best guess”) contaminant concentrations from data with censored values (e.g., less than the detection limit). The method represents an advancement over existing techniques because it is capable of estimating the proportion of measurements that are true zeros and incorporating varying levels of censorship (e.g., sample specific detection limits, changes through time in method detection). The ability of the method to estimate the proportion of true zeros is validated using precipitation data. The stability and flexibility of the method are demonstrated with stochastic simulation, a sensitivity analysis, and unbiasedness analysis including varying numbers of significant digits. A key aspect of this paper is the application of the statistical analysis to real site rock core contaminant concentration data collected within a plume at two locations using high resolution depth-discrete sampling. Comparison of the representative values for concentrations at each location along the plume center-line shows a larger number of true zeros and generally lower concentrations at the downgradient location according to the conceptual site model, leading to improved estimates of attenuation with distance and/or time and associated confidence; this is not achievable using deterministic methods. The practical relevance of the proposed method is that it provides an improved basis for evaluating change (spatial, temporal, or both) in environmental systems.

Zusammenfassung

This paper demonstrates quantitative reasoning to separate the dataset of spatially distributed variables into different entities and subsequently characterize their geostatistical properties, properly. The main contribution of the paper is a statistical based algorithm that matches the manual distinction results. This algorithm is based on measured data and is generally applicable. In this paper, it is successfully applied at two datasets of saturated hydraulic conductivity (K) measured at the Borden (Canada) and the Lauswiesen (Germany) aquifers. The boundary layer was successfully delineated at Borden despite its only mild heterogeneity and only small statistical differences between the divided units. The methods are verified with the more heterogeneous Lauswiesen aquifer K data-set, where a boundary layer has previously been delineated. The effects of the macro- and the microstructure on solute transport behaviour are evaluated using numerical solute tracer experiments. Within the microscale structure, both Gaussian and non-Gaussian models of spatial dependence of K are evaluated. The effects of heterogeneity both on the macro- and the microscale are analysed using numerical tracer experiments based on four scenarios: including or not including the macroscale structures and optimally fitting a Gaussian or a non-Gaussian model for the spatial dependence in the micro-structure. The paper shows that both micro- and macro-scale structures are important, as in each of the four possible geostatistical scenarios solute transport behaviour differs meaningfully.

Zusammenfassung

Abstract For the reconstruction and interpolation of precipitation fields, we present the application of a stochastic approach called Random Mixing. Generated fields are based on a data set consisting of rain gauge observations and path-averaged rain rates estimated using Commercial Microwave Link (CML) derived information. Precipitation fields are received as linear combination of unconditional spatial random fields, where the spatial dependence structure is described by copulas. The weights of the linear combination are optimized such that the observations and the spatial structure of the precipitation observations are reproduced. The innovation of the approach is that this strategy enables the simulation of ensembles of precipitation fields of any size. Each ensemble member is in concordance with the observed path-averaged CML derived rain rates and additionally reflects the observed rainfall variability along the CML paths. The ensemble spread allows additionally an estimation of the uncertainty of the reconstructed precipitation fields. The method is demonstrated both for a synthetic data set and a real-world data set in South Germany. While the synthetic example allows an evaluation against a known reference, the second example demonstrates the applicability for real-world observations. Generated precipitation fields of both examples reproduce the spatial precipitation pattern in good quality. A performance evaluation of Random Mixing compared to Ordinary Kriging demonstrates an improvement of the reconstruction of the observed spatial variability. Random Mixing is concluded to be a beneficial new approach for the provision of precipitation fields and ensembles of them, in particular when different measurement types are combined.

Zusammenfassung

The geostatistical workflow of data analysis, model fitting, and subsequent interpolation or simulation has recently been enhanced by several methods. These methods can be summarized under the terms ’rank-order geostatistics’, for the empirical analysis part of the workflow, and ’copula geostatistics’, for the theoretical foundation and the modeling (interpolation or simulation). Besides taking into account non-Gaussianity, the main advantage of this alternative way to treat geostatistical problems is the descriptive power and standardized interpretability. This paper addresses the empirical analysis part of the workflow. We investigate the interplay between structural features, statistical properties, and underlying dynamic processes of a realization of a spatial field. (1) In the first part of this paper we recapitulate and consolidate the advances in the empirical analysis part of the geostatistical workflow and put them into context of ’classical’ geostatistics. We place particular emphasis on the theoretical foundation of so called asymmetry functions, because in our opinion they are the necessary first step away from Gaussian geostatistics. (2) In the second part, we rigorously analyze how specific types of structural features are related to the asymmetry of a spatial field. In the geostatistical tradition of interpreting the shape of variograms or correlation functions, we give examples of how to interpret different shapes of the asymmetry function. (3) We subsequently report how dynamic processes naturally lead to the emergence of asymmetrical (non-Gaussian) spatial structures. To demonstrate these findings, we investigate the manifestation of different spatial data sets (land surface elevation, groundwater contamination, grades of an undergraduate exam) and relate the non-Gaussian structural features to the underlying dynamic processes. Numerical process models are utilized to manifest evidence of how realistic processes naturally lead to complex non-Gaussian structures. The key purpose of this paper is to show that asymmetry is a fundamental geostatistical property and a result of different kinds of spatial processes.

Zusammenfassung

The use of radar measurements for the space time estimation of precipitation has for many decades been a central topic in hydro-meteorology. In this paper we are interested specifically in daily and sub-daily extreme values of precipitation at gauged or ungauged locations which are important for design. The purpose of the paper is to develop a methodology to combine daily precipitation observations and radar measurements to estimate sub-daily extremes at point locations. Radar data corrected using precipitation-reflectivity relationships lead to biased estimations of extremes. Different possibilities of correcting systematic errors using the daily observations are investigated. Observed gauged daily amounts are interpolated to unsampled points and subsequently disaggregated using the sub-daily values obtained by the radar. Different corrections based on the spatial variability and the subdaily entropy of scaled rainfall distributions are used to provide unbiased corrections of short duration extremes.Additionally a statistical procedure not based on a matching day by day correction is tested. In this last procedure as we are only interested in rare extremes, low to medium values of rainfall depth were neglected leaving a small number of L days of ranked daily maxima in each set per year, whose sum typically comprises about 50% of each annual rainfall total. The sum of these L day maxima is first iterpolated using a Kriging procedure. Subsequently this sum is disaggregated to daily values using a nearest neighbour procedure. The daily sums are then disaggregated by using the relative values of the biggest L radar based days. Of course, the timings of radar and gauge maxima can be different, so the method presented here uses radar for disaggregating daily gauge totals down to 15 min intervals in order to extract the maxima of sub-hourly through to daily rainfall.The methodologies were tested in South Africa, where an S-band radar operated relatively continuously at Bethlehem from 1998 to 2003, whose scan at 1.5 km above ground CAPPI overlapped a dense (10 km spacing) set of 45 pluviometers recording in the same 6-year period. This valuable set of data was obtained from each of 37 selected radar pixels 1 km square in plan which contained a pluviometer not masked out by the radar foot-print. The pluviometer data were also aggregated to daily totals, for the same purpose. The extremes obtained using disaggregation methods were compared to the observed extremes in a cross validation procedure. The unusual and novel goal was not to obtain the reproduction of the precipitation matching in space and time, but to obtain frequency distributions of the point extremes, which we found to be stable.

Zusammenfassung

Natural processes generate spatial fields which reflect their specific properties. In this paper the effect of the direction of processes on the resulting spatial fields is investigated. This is done by extending the concept of reversibility used for time series to space. A novel copula based measure of asymmetry is defined which is an indicator of directional dependence. Contrary to traditional geostatistics where all points separated by a vector are considered irrespective of its sign, in this study the direction of the vector is also taken into consideration, leading to differences in the dependence corresponding to the vector \$\$\\backslashmathbf \h\\\$\$hand \$\$\-\\\backslashmathbf \h\\\$\$-h. The concept of directional dependence and the corresponding measure of asymmetry are defined using spatial copulas, and are thus independent of the scale of measurement. The result is a bivariate directional third-order moment based measure which can identify the direction in which the processes generating the spatial field acted. A statistical test to find the statistical significance of the asymmetry indicating directional dependence is presented. The methodology is tested on a number of synthetic and observed cases. Precipitation and groundwater quality parameters obtained using numerical models are first investigated. Regular dense grids obtained by numerical simulations show good correspondence between properties of the modeled processes and the new measure introduced. Measured variables observed on sparse irregular networks show similar behavior to the theoretical examples. Mean flow directions in groundwater and advection directions of precipitation fields can be detected from single snapshots. As a further example, dominant wind directions in the Sahara are found by investigating the digital terrain model.